Ergodic properties of heterogeneous diffusion processes in a potential well

Abstract

Heterogeneous diffusion processes can be well described by an overdamped Langevin equation with space-dependent diffusivity $D(x)$. We investigate the ergodic and non-ergodic behavior of these processes in an arbitrary potential well $U(x)$ in terms of the observable---occupation time. Since our main concern is the large-$x$ behavior for long times, the diffusivity and potential are, respectively, assumed as the power-law forms $D(x)=D_0|x{|}^{\alpha }$ and $U(x)=U_0|x{|}^{\beta }$ for simplicity. Based on the competition roles played by $D(x)$ and $U(x)$, three different cases, $\beta >\alpha$, $\beta =\alpha$, and $\beta <\alpha$, are discussed. The system is ergodic for the first case $\beta >\alpha$, where the time average agrees with the ensemble average, being both determined by the steady solution for long times. In contrast, the system is non-ergodic for $\beta <\alpha$, where the relation between time average and ensemble average is uncovered by infinite-ergodic theory. For the middle case $\beta =\alpha$, the ergodic property, depending on the prefactors $D_0$ and $U_0$, becomes more delicate. The probability density distribution of the time averaged occupation time for three different cases are also evaluated from Monte Carlo simulations.

14 pages, 9 figures

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